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Homework Help: Partial fraction (tricky stuff for LaPlace transforms)

  1. Jul 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Doing some Laplace transform stuffs and I've got [tex]Y(s) = \frac{1}{(s+1)(s^{2}+2s+2)}[/tex]

    Using the normal method

    [tex]1 = A((s+1)^{2}+1) + B(s+1)[/tex]

    I'm not sure this method is valid though as we had a complicated term (s+1)² + 1 after A. I can find A to be 1, but I don't trust my value of B being -1.

    Apparently the answer is [tex]\frac{1}{s+1} - \frac{s+1}{(s+1)^{2}+1}[/tex]

    Not sure how to get there though

    Thanks
    Thomas
     
  2. jcsd
  3. Jul 28, 2010 #2
    The discrimant of the quadratic trinomial is:

    [tex]
    \Delta = 1^{2} - 1 \cdot 2 = 1 - 2 = -1 < 0
    [/tex]

    so it does not have real zeros. That is why the partial fraction expansion is of the form:

    [tex]
    \frac{1}{(s + 1)(s^{2} + 2 s + 2)} = \frac{A}{s + 1} + \frac{B s + C}{s^{2} + 2 s + 2}
    [/tex]

    Multiply both sides by [itex](s + 1)(s^{2} + 2 s + 2)[/itex], collect terms with like powers of s on the right hand side and compare the corresponding coefficients so that you will get three equations for the three unknowns A, B and C. Also, complete the square for the trinomial:

    [tex]
    s^{2} + 2 s + 2
    [/tex]

    You should get the answer you were looking for.
     
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